AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 123:156 –171 (2004)
Predicting Long Bone Loading From Cross-Sectional
Geometry
Daniel E. Lieberman,1* John D. Polk,1 and Brigitte Demes2
1
2
Department of Anthropology, Peabody Museum, Harvard University, Cambridge, Massachusetts 02138
Department of Anatomical Sciences, Stony Brook University, Stony Brook, New York 11794-8081
KEY WORDS
second moment of area; cross-sectional geometry; in vivo strain; sheep;
functional morphology
ABSTRACT
Long bone loading histories are commonly evaluated using a beam model by calculating crosssectional second moments of areas (SMAs). Without in
vivo strain data, SMA analyses commonly make two explicit or implicit assumptions. First, while it has long been
known that axial compression superimposed on bending
shifts neutral axes away from cross-sectional area centroids, most analyses assume that cross-sectional properties calculated through the area centroid approximate
cross-sectional strength. Second, the orientation of maximum bending rigidity is often assumed to reflect the orientation of peak or habitual bending forces the bone experiences. These assumptions are tested in sheep in which
rosette strain gauges mounted at three locations around
the tibia and metatarsal midshafts measured in vivo
strains during treadmill running at 1.5 m/sec. Calculated
normal strain distributions confirm that the neutral axis
of bending does not run through the midshaft centroid. In
these animals, orientations of the principal centroidal
axes around which maximum SMAs (Imax) are calculated
are not in the same planes in which the bones experienced
bending. Cross-sectional properties calculated using centroidal axes have substantial differences in magnitude (up
to 55%) but high correlations in pattern compared to crosssectional properties calculated around experimentally determined neutral axes. Thus interindividual comparisons
of cross-sectional properties calculated from centroidal
axes may be useful in terms of pattern, but are subject to
high errors in terms of absolute values. In addition, crosssectional properties do not necessarily provide reliable
data on the orientations of loads to which bones are subjected. Am J Phys Anthropol 123:156 –171, 2004.
For many years, cross-sectional geometrical properties have been used to test hypotheses and make
inferences about the mechanical adaptations of long
bones. Many early studies of long bones examined
cross-sectional area (CSA), which reflects a bone’s
resistance to axial compression and tension. More
recently, second moments of area (SMAs, also
termed area moments of inertia) were used preferentially in analyses of cross-sectional geometry because they better characterize the resistance of a
long bone to bending around a particular axis (Wainright et al., 1976). Measures of CSA or bone thickness can provide poor estimates of strength to nonaxial loading because resistance to bending depends
on the distribution of area in a cross section. SMA
(designated I) measures how material is distributed
about a defined axis as:
tance, y, from the neutral axis of bending (NA)
(Fig. 1):
I ⫽
冕
⫹y max
y 2 dA
⫺y max
where A is area and y is the perpendicular distance
to an axis through the centroid. In nonintegral form,
I can be approximated by dividing a section into
many squares of dimension x2, each located a dis©
2004 WILEY-LISS, INC.
©
2004 Wiley-Liss, Inc.
冘 x 䡠y
n
I ⫽
2
i
2
i
i
Typically, I is calculated with respect to orthogonal
axes defined by anatomical or arbitrary reference
planes (Ixx, Iyy), or I may be calculated with respect
to orthogonal principal axes (Imax, Imin). I, in turn,
permits one to calculate the section modulus, Z,
defined as I ● (D ● 0.5)⫺1, where D is the section’s
diameter in the plane of bending. Z best approximates a cross section’s resistance to bending in a
given plane (Wainright et al., 1976). Polar moments
Grant sponsor: NSF; Grant number: IBN 96-03833.
*Correspondence to: Daniel E. Lieberman, Department of Anthropology, Peabody Museum, Harvard University, Cambridge, MA
02138.
Received 30 September 2002; accepted 22 March 2003.
DOI 10.1002/ajpa.10316
Published online 10 June 2003 in Wiley InterScience (www.
interscience.wiley.com).
LONG BONE LOADING
Fig. 1. Idealized cylindrical beam subject to loading at proximal end from body mass and at distal end from ground reaction
force. Axial forces (Fa), which generate compression, can be resolved into transverse (Ft) and vertical (Fv) components. NA,
neutral axis around which compressive (⫹) and tensile (⫺)
strains occur. Bending moment at midshaft cross section is counteracted by second moment of area, I. See text for further details.
of area (or polar second moments of inertia), J, are
calculated as the sum of any two perpendicular measures of I (e.g., Ixx ⫹ Iyy or Imax ⫹ Imin); J describes
both resistance to torsion of circular cross sections
(Wainright et al., 1976; Carter and Beaupré, 2001;
Daegling, 2002) and average bending rigidity
(Schaffler et al., 1985). In addition, ratios of orthogonal SMAs (Ixx ● Iyy⫺1 or Imax ● Imin⫺1) were hypothesized to be useful quantifiers of cross-sectional
shape, potentially providing information on the type
of loading to which a bone is adapted and/or subjected (Lovejoy et al., 1976; Ruff and Hayes, 1983).
Limb bone cross-sectional properties have widely
been used to make inferences about long bone adaptations in all sorts of vertebrates (e.g., Alexander,
1983; Selker and Carter, 1989; Biknevicius, 1993;
Heinrich and Rose 1997; Cubo and Casinos, 1998;
Heinrich et al., 1999), including comparisons between human populations (e.g., Ruff and Hayes,
1983; Ruff and Larsen, 1990; Ruff, 1992; van der
Meulen et al., 1993; Bridges et al., 2000; Stock and
Pfeiffer, 2001), between hominid species (Trinkaus,
1997; Trinkaus and Churchill, 1999; Trinkaus and
Ruff, 1999; Trinkaus et al., 1999; Churchill and
Smith, 2000), and between primate species (e.g.,
Schaffler et al., 1985; Ruff, 1987; Burr et al., 1989;
Demes and Jungers, 1989, 1993; Demes et al., 1991;
Ruff and Runestad, 1992; Ohman, 1993; Jungers
157
and Burr, 1994; Terranova, 1995; Runestad, 1997;
Jungers et al., 1998; Polk et al., 2000; Llorens et al.,
2001; Kimura, 2002). In such comparisons, crosssectional properties standardized by body mass
and/or element length or dimensionless ratios such
as Imax ● Imin⫺1 provided important insights about
loading history and locomotor adaptations. For example, it was long thought that Neanderthals had
more robust bones than modern humans (e.g.,
Trinkaus, 1992). However, more recent analyses
that plotted cross-sectional properties against tibial
and femoral lengths adjusted for body mass and
shape (for details of this adjustment, see Ruff et al.,
1993; Trinkaus, 1997) found no significant difference in adjusted J between Neanderthals and early
“anatomically modern” humans from the Near East,
but significant differences in cross-sectional shape
(Ixx ● Iyy⫺1) (Trinkaus et al., 1998a, b; Trinkaus and
Ruff, 1999). As a result, these authors concluded
that Neanderthals and early “anatomically modern”
modern humans had similar magnitudes but possibly different patterns of locomotor loading.
Several factors, however, complicate the use of
cross-sectional properties for testing hypotheses
about bone function or loading history. Most importantly, calculating cross-sectional properties from
bone midshafts requires one either to determine experimentally or to make some assumptions about
how a bone is bent during loading, using a beam
model. Beams are defined as structural elements
with fairly straight long axes whose length is several
times greater than their width and depth. In general, long bones are typically modeled as solid or
hollow cylindrical beams subjected to two-point
bending by forces applied to their ends, as seen in
Figure 1. Under such conditions, bending stress/
strain at any location is a function of the magnitude
and orientation of the two forces and the cross-sectional and material properties of the bone. For example, limb bones are often modeled at midstance,
when ground reaction forces (GRFs) are high and
approximately vertical. Elements not aligned with
the GRF will be bent so that anterior and posterior
cortices will experience compression or tension, respectively, depending on whether the GRF passes in
front of or behind the element. In the case of the
typical quadruped leg, illustrated in Figure 2A, the
cranial (anterior) cortex of the metatarsal is predicted to be in compression, and the cranial cortex of
the tibia is predicted to be in tension. Note also that
more angled elements such as the tibia should experience greater bending stresses/strains because
the moment arm to the GRF is longer (Biewener et
al., 1983).
If a long bone is loaded in pure bending, then the
neutral axis of bending will run through the area
centroid of the shaft section (Wainright et al., 1976).
However, it has long been appreciated that long
bones are loaded by a combination of bending moments and axial forces (e.g., Pauwels, 1948; Lovejoy
et al., 1976). Under such conditions, the neutral axis
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D.E. LIEBERMAN ET AL.
Fig. 2. A: Schematic view of sheep hindlimb at midstance. At
right, approximate orientation of ground reaction force (Fg), relative to orientation of metatarsal and tibia. Cranial (anterior)
cortex of metatarsal is predicted to be compressed. Bending in
tibia depends on where it intersects with Fg (i.e., cranial part of
proximal shaft should be tensed). B: Effect of axial compression
on neutral axis position. At left, a section subject to pure bending,
with hypothetical values of tension (positive) and compression
(negative). Superimposition of an evenly distributed compressive
force of ⫺1, causes neutral axis (NA) to shift towards cortex,
subject to tension.
will not pass through the centroid; instead, the superimposition of bending and axial compression
shifts the neutral axis (NA) of bending away from
the centroidal axis (CA) towards the cortex under
tension, as illustrated in Figure 2B. Such shifts of
the NA were demonstrated experimentally in the
few published studies in which strain gauges were
applied to at least three locations around a bone
midshaft to solve the location of the neutral axis
(Carter et al., 1981; Szivek et al., 1992; Gross et al.,
1992; Judex et al., 1997; Demes et al., 1998, 2001).
In addition, a few experiments bonded strain gauges
to metal plates affixed to long bones (Carter et al.,
1981; Gautier et al., 2000), with similar results.
Unfortunately, such experiments are difficult to perform on most bones in most species, because the
application of three gauges around a bone midshaft
is liable to affect muscle function adversely, causing
lameness or other abnormal kinematic and kinetic
patterns. In addition, no such data are available for
humans because, so far, the only way to estimate NA
position is to apply strain gauges in vivo.
In the absence of experimental data on NA position (as is the case for all human studies), most
researchers calculated cross-sectional geometric
properties using the area centroid. This approach
requires the assumption (sometimes explicit, but
usually implicit) that there is a predictable, reliable
relationship between cross-sectional geometric properties calculated around the CA vs. those calculated
around the NA. Most analyses of cross-sectional geometry do not assume the absolute values of crosssectional geometric properties to be correct, but the
relative values or patterns of cross-sectional geometric
properties are assumed to be useful if not approximately correct for making comparisons between individuals or species. This assumption has mostly been
made for J, which, as the sum of any two orthogonal
measures of I, “provides an indication of overall biomechanical structural integrity” (Trinkaus, 1997, p.
13,368). However, this assumption has never been
tested by calculating cross-sectional geometric properties using experimentally determined neutral axes.
A second major problem is the relationship between form and function in the midshaft. In order to
make inferences about loading history or structural
adaptations in bones using cross-sectional properties, there has to be some predictable relationship
between the orientation of bending in a bone and the
distribution of area within the cross section. While J
is often considered to be a general indicator of bending strength (see above), interpretations of other
cross-sectional geometric parameters (notably Imax,
Imin, and Z) are potentially more problematic. In
particular, since Imax indicates the relative magnitude of greatest bending rigidity in a section, it has
been considered reasonable to hypothesize that the
orientation of bending is approximated by Imax. As
stated by Ruff and Hayes (1983, p. 379), “Bone area
appears to be distributed in such a way as to minimize stresses and strains developed under loading.”
However, experimental studies of limb shaft biomechanics yielded a confusing picture of the relationship between midshaft shape and cross-sectional
distribution of strains. The analysis by Szivek et al.
(1992) of femoral strain in the greyhound showed
that the orientation of the NA was nearly anteroposterior at 0.6 m/sec (a slow walk), but rotated to 45°
relative to a mediolateral axis at 1.2 m/sec (a slow
trot), and to within 15° relative to a mediolateral
axis at 2.4 m/sec (a moderate trot), when strains
were higher. Gross et al. (1992) showed that the
mediolateral orientation of the NA in the horse third
metacarpal bone indicates anteroposterior bending
at 3.6 m/sec (a moderate trot), even though the wide
shape of the third metacarpal midshaft is clearly
strongest to withstand mediolateral bending. Similarly, in studies by Demes et al. (1998, 2001) of the
macaque tibia and ulna, the orientations of the NA
differed substantially from Imax in both walking and
galloping. At a walk, the macaque tibia was bent
obliquely between 30 – 45° relative to Imax, and the
macaque ulna was bent mediolaterally, approximately 90° relative to Imax. At higher speeds (a gallop), the NA of the tibia became more mediolaterally
oriented (by about 20°). Thus, the extent to which
loading patterns can be inferred from bone crosssectional shape is not clear. Previous researchers
LONG BONE LOADING
who noted the differences between cross-sectional
bone shape and loading, mostly in distal elements,
suggested that seemingly nonoptimal cross-sectional shapes are adaptations to increase the predictability of loads at the expense of strength
(Lanyon and Rubin, 1985; Bertram and Biewener,
1988). However, with the exception of Demes et al.
(2001), none of these studies explicitly tested the
relationship between the orientations of NA and
Imax.
Finally, dynamic changes in loads applied to
bones during the stance phase at different speeds or
gaits could further complicate the relationship between cross-sectional geometric properties and how
bone shafts are bent (Biewener et al., 1988). Most of
the experimental studies cited above, that calculated cross-sectional gradients using in vivo strain
data, examined just midstance, when peak strains
are typically highest. Only a few studies looked at
dynamic changes through gait, with slightly different results. Demes et al. (1998, 2001) found no major
change in how the macaque ulna and tibia were bent
during the stance phase. Szivek et al. (1992) found
little change in the orientation of NA in the dog
femur between midstance and toe-off, but about 15°
rotation between paw strike and midstance. Thus
additional evidence is needed to characterize how
bending regimes change during the stance phase. To
complicate matters, midstance loading patterns
sometimes differ by speed and gait. As noted above,
Szivek et al. (1992) found more anteroposterior
bending in the dog femur at higher speeds. Thus the
shape of the dog femoral midshaft appears to reflect
the orientation of forces applied to the bone at
higher speeds. Demes et al. (2001), however, showed
that the orientation of bending in the tibia, but not
the ulna, in macaques shifted to a more anteroposterior pattern of bending between walking and galloping.
The above problems raise several questions about
the relationship between bone cross-sectional shape
and how bone shafts are loaded. First, to what extent is the direction of bending determined by bone
shape, by local muscles forces, or by the ground
reaction force and its position relative to the shaft?
While in vivo strain data offer some support for the
hypothesis that ground reaction force orientation
relative to limb orientation is the primary determinant of bending orientation (Biewener, 1982, 1983;
Biewener et al., 1983), these data were experimentally determined for only a few animals and limbs,
and obviously cannot be known for extinct species.
In addition, using the ground reaction force vector
projected up the limb to assess bending moments is
a method prone to error, especially for more proximal segments, where the discrepancy between the
ground reaction force moment and the actual joint
moment applied to the bone can be considerable
(Wells, 1981; Winter, 1990). A second question is,
how much do muscle and tendon forces influence
patterns of bone loading? One in vitro study sug-
159
gests that muscles and tendons may have some effects on overall bone loading (Peterman et al., 2001),
and several experimental studies show that muscles
contribute substantially to local stresses and strains
(Barnes and Pinder, 1974; Biewener et al., 1992;
Dial and Biewener, 1993). But more research is
needed to test the effects of muscle forces on midshaft strains. Such forces, for example, were modeled but not tested in vivo with respect to gluteal
abductor forces on the shaft of the femoral neck
(Lovejoy et al., 1973; Ruff, 1995). And finally, what
is the relationship between bone loading and midshaft shape? Application of Wolff’s law to cross-sectional geometry was experimentally tested only partially (see also Biewener et al., 1986; Ruff et al.,
1993, 1994; Lieberman, 1996; Qin et al., 1998; Goodship and Cunningham, 2001; Lieberman and Pearson, 2001; Rubin et al., 2001), and requires more
research to examine the interactions of loading parameters, age, and skeletal location on phenotypic
plasticity of cross-sectional geometry. While many
studies showed that loading can stimulate growth
(for a review, see Martin et al., 1999; but see also
Bertram and Swartz, 1991), few experiments directly related orientations of experimentally measured bending to patterns of growth. Judex et al.
(1997) found that rates of periosteal growth in the
tarsometatarsus of adult roosters (which was mostly
bent anteroposteriorly) correlated poorly with sites
of peak strain magnitude. In contrast, Gross et al.
(1997) found a rough correspondence between sites
of peak compression and sites of peak bone growth
in turkey radii that were functionally isolated and
then artificially loaded. More studies are needed to
evaluate growth changes in cross-sectional geometric properties in relation to experimentally determined midshaft cross-sectional strains.
HYPOTHESES TO BE TESTED
As noted above, many problems remain to be addressed in order to improve our understanding of the
relationships between bone cross-sectional shape,
bone function, and bone loading history. We focus
here on the relationship between patterns of loading
and cross-sectional geometry by using rosette strain
gauges mounted at three sites around the midshaft
of the tibia and metatarsal in juvenile sheep. By
solving for the magnitude of strain normal to the
long axis of the midshaft (see below) from each
strain gauge, the position of the neutral axis and the
cross-sectional distribution of tensile and compressive strains can be calculated based on simple beam
bending theory (Rybicki and Mills, 1977). By analyzing the cross-sectional geometry of the same midshafts, we tested three hypotheses. First, while it
has long been appreciated that the NA does not
necessarily run through the area centroid of a midshaft, it was commonly assumed either explicitly or
implicitly that measurements of cross-sectional geometric properties calculated using centroidal axes
approximate, either in pattern or magnitude, cross-
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D.E. LIEBERMAN ET AL.
sectional geometric properties that are calculated
through an experimentally determined NA. We
therefore tested the hypothesis that cross-sectional
geometric properties calculated around an experimentally determined NA are equal to or approximate those calculated through the CA. Second, it is
commonly assumed (sometimes explicitly but often
implicitly) that the distribution of area in a midshaft
cross section is related to the pattern of loading to
which the bone is subjected. We therefore tested the
hypothesis that the NA coincides in its orientation
with the principal, centroidal axis of the cross section (CAmax, the axis around which the cross section
offers greatest resistance to bending). Third, there
are conflicting studies of the relationship between
the distribution of peak strains in the midshaft and
the dynamic changes in bending that occur throughout stance. We therefore tested the hypothesis that
the orientation and position of the NA during the
stance phase are stable and similar to its position at
midstance.
In addition to these hypotheses, we also attempted to explore the statistical relationship between cross-sectional geometric parameters (I, J,
and Z) calculated using CAs vs. NAs by extrapolating the results of in vivo strain gauge analyses to
cross-sectional geometric properties in a large sample of sheep that were exercised on a treadmill for 3
months at the same speed for which we measured
strains.
MATERIALS AND METHODS
Subjects
Dorset sheep (Ovis aries) were used for this experiment because they are docile, good treadmill runners, and have relatively little muscle mass on the
tibia and metatarsal, permitting application of
strain gauges at multiple sites without impairing
normal gait. Strain gauges were applied to 5 juvenile sheep, all between 20 –30 kg, who were trained
to run at speeds up to 3 m/sec on a treadmill. Data
presented here are for strains obtained at 1.5 m/sec
(a moderate trot). For comparison, we also examined
data on cross-sectional properties, element lengths,
and body mass from Dorset sheep from three age
categories: juvenile (aged 110 days, n ⫽ 10),
subadult (aged 355 days, n ⫽ 10), and young adult
(aged 505 days, n ⫽ 16); half of these sheep had been
exercised daily for 90 days at approximately 1.5
m/sec (for additional details of this sample, see
Lieberman et al., 2001).
Strain data acquisition
Prior to surgery, animals used in strain gauge
experiments were sedated with ketamine (8.0 mg/
kg, i.m.), xylazine (0.05 mg/kg, i.m.), and atropine
(0.05 mg/kg, i.m.), intubated, and maintained on a
surgical plane of anesthesia with isoflurane. The left
hindlimb of each animal was shaved and sterilized,
and the location of the midshaft was marked. Under
sterile surgical conditions, insulated FRA-1-11 rosette strain gauges (Sokki Kenkyujo, Tokyo, Japan)
with 120 ⫾ 0.5 Ohm resistance were affixed to the
anterior, medial, and posterior surfaces of the tibial
midshaft through an incision on the medial surface,
and to the anterior, medial, and lateral surfaces of
the metatarsal midshaft from incisions on the medial and lateral surfaces of the leg. Gauges were
sealed using M-coat and D-coat (MicroMeasurements, Inc., Raleigh, NC). To provide anesthesia and
minimize inflammation, bupivacaine (diluted 1:10)
was injected subcutaneously around each incision
site. Muscles and tendons were retracted on the
posterior and anterior surface of both bones during
gauge insertion, but care was taken to ensure that
these structures were not damaged. The surface of
the bone at each gauge site was prepared by cutting
a small window (ca. 5 mm2) in the periosteum, cauterizing any vessels, and degreasing with 100% chloroform. Bupivacaine (diluted 1:10) was perfused under the periosteum prior to cutting, to provide
anesthesia. Gauges were bonded using methyl-2cyano-acrylate glue, with continuous pressure applied for 2 min as the glue was drying. Care was
taken to ensure that one of the elements of the gauge
was aligned with the long axis of the bone. The
orientations of each gauge’s A-element (previously
marked on the gauge’s sealing coat using metallic
ink) relative to the long axis of the bone were recorded prior to closing the incision with suture.
Gauge leads were then passed extracutaneously underneath flexible bandages to the hip, where they
were sutured to a bandage loosely wrapped around
the animal’s abdomen. To provide strain relief, the
leads of each gauge were affixed to a bandage
wrapped around the leg near the site of the incision.
Strain data were recorded approximately 4 and 24
hr after surgery. Recordings were made when animals were running with an apparently normal gait
and showed no signs of lameness, distress, or discomfort (e.g., with symmetrical limb kinematics on
the operated and nonoperated hindlimbs, and no
signs of leaning or favoring one limb over another).
During each recording session, gauges were connected with insulated wire to Vishay 2120A amplifiers (MicroMeasurements, Inc., Raleigh, NC) to
form one arm of a Wheatstone quarter-bridge circuit; bridge excitation was 1 V. Voltage outputs were
recorded on a Teac RD-145T DAT tape recorder
(Teac Corp., Tokyo, Japan). Gauges were periodically balanced to adjust for zero offsets during the
experiment, and calibrated when the animal was
stationary and the instrumented limb was off the
ground.
To correlate strain data with limb kinematics,
three-dimensional (3D) coordinates were obtained
for all hindlimb joints, using a three-camera Qualisys (Qualisys, Inc., Gothenburg, Sweden) infrared
3D motion analysis system. This system tracked the
position of reflective markers (12 mm in diameter)
placed on the shaved skin overlying the distal inter-
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LONG BONE LOADING
phalangeal joint, distal metatarsal, lateral malleolus, lateral epicondyle of the femur, greater trochanter, and anterior superior iliac spine. Capture
frequency was set at 60 Hz, and kinematic sequences were synchronized to output from the strain
gauges, using a trigger that started data capture by
the Qualysis system at the same time that a 2-V
pulse signal was sent to the tape recorder. Limb
segments were identified by connecting adjacent
markers. QTools software (Qualisys, Inc.) was used
to identify and measure element orientation at different points in the stance cycle, including touchdown, 25% through stance phase, temporal midstance, and 75% through stance phase.
Strain analysis
Selected sequences of strain data were sampled
from the tape recorder on a Macintosh G4 computer,
using an Ionet™ A-D board (GW Instruments, Somerville MA) at 250 Hz. A Superscope 3.0™ (GW
Instruments) virtual instrument (written by D.E.L.)
was used to determine the zero offset, calculate
strains (in microstrain units (␮⑀) of principal tension
(⑀1) and compression (⑀2) from raw voltage data using shunt calibration signals recorded during the
experiments, and calculate the orientation of principal tension in degrees relative to the axis of the long
axis of the limb (⑀1), using formulae from Biewener
(1992). In addition, a Mohr’s circle analysis was used
to calculate the magnitude of strain normal to the
cross section (⑀y) for each gauge, following formulae
in Hibbeler (1999). Igor Pro version 3.01 (Wavemetrics, Inc., Lake Oswego, OR) was used to analyze
strain data from different points in the stance cycle
(see above) for at least 10 stance phases per animal.
To characterize midshaft strain environment, digitized transverse cross sections of each bone midshaft were analyzed with an NIH Image macro
(written by S. Martin, University of Melbourne) to
calculate and graph the neutral axis and gradients
of normal strain across the section, using a beam
analysis model (formulae in Rybicki et al., 1974;
Biewener, 1992; Gross et al., 1992). The macro
solves three linear equations:
␧ 1 ⫽ ax 1 ⫹ by 1 ⫹ c
␧ 2 ⫽ ax 2 ⫹ by 2 ⫹ c
␧ 3 ⫽ ax 3 ⫹ by 3 ⫹ c
where ⑀1, ⑀2, and ⑀3 are the normal strains recorded
at three sites on the bone cortex with the coordinates
(x1, y1), (x2, y2), and (x3, y3), and a, b, and c are
constants. The longitudinal normal strains measured by the three strain gauges (⑀1 through ⑀3) and
their respective coordinate locations (x1–3, y1–3) are
used to solve for the three unknown parameters.
The equation defining any strain isocline, ⑀i, including the neutral axis (where ⑀i ⫽ 0), can then be
solved as:
␧ i ⫽ ax ⫹ by ⫹ c,
and x and y coordinates of points on the neutral axis
can be calculated using this equation.
It is not possible to calculate error for strain gauge
isoclines. However, strain gauges provide highly accurate and precise data when they are bonded
tightly to the bone surface and calibrated properly
(for discussion, see Biewener, 1992). Slight errors
most commonly occur from temporal fluctuations in
the balance of the Wheatstone quarter bridges, and
from inaccurate measurements of the position of the
gauges on the midshaft cross section or the orientation of the gauge’s elements relative to the long axis
of the bone. Gauge positions and orientations, however, were measured ex vivo. In addition (as noted
above), gauges were frequently balanced during the
experiment, with adjustments typically less than 5
mV (1 mV is equivalent to about 1␮⑀).
Morphological measurements
Following euthanasia of animals used in strain
gauge analysis, the hindlimbs were dissected, and
the orientation of each gauge was recorded. In all
animals, biarticular lengths of the femur, tibia, and
metatarsal were measured using digital calipers.
Femoral length was measured from the most proximal point on the femoral head to the line connecting
the two distal condyles; tibial length was measured
from the center of the lateral condylar surface to the
center of the distal articular surface; metatarsal
length was measured from the center of the proximal articular surface to the most distal point of the
distal articular surface. A 1-cm section was then cut
from the midshaft of the tibia and metatarsal at the
level of the gauges, and photographed using a digital
camera on 1-mm graph paper.
SMA calculations
Cross-sectional properties were calculated in two
ways. First, an NIH Image version 1.61 (http://rsb.
info.nih.gov/nih-image/) macro written by M. Warfel
(Cornell University) was used to calculate SMAs,
using the assumption that the NA runs through the
area centroid of each cross section. The macro works
by calculating the coordinates of the area centroid,
and then calculating I as the sum of the areas of
each pixel times their squared distances to defined
axes that pass through the centroid. Ixx and Iyy are
area moments calculated around centroidal axes
(CA) in the mediolateral (CAxx) and anteroposterior
(CAyy) planes; Imax and Imin are the area moments
around the principal axes (CAmax, CAmin). Jc was
calculated as Imax ⫹ Imin (which equals Ixx ⫹ Iyy);
and Zc was calculated for the anteroposterior axis
(following the common assumption that limb bones
are bent in the sagittal plane).
Second, a different NIH macro (written by S. Martin, University of Melbourne) was used to calculate
SMA around the experimentally determined neutral
axis (see above), defined here as IN. INy is calculated
(somewhat arbitrarily) around the axis orthogonal
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D.E. LIEBERMAN ET AL.
TABLE 1. Tibia midstance strain data (1.5 m/sec, 10 strides)1
Posterior cortex
Subject
Normal strain
616
574
562
539
600
Grand mean
⫺428.5 (51.3)
Medial cortex
ε1 angle
Normal strain
ε1 angle
⫺76.4 (0.9)
⫺76.4 (na)
⫺215.2 (46.4)
⫺449.4 (71.8)
⫺312.4 (52.4)
⫺191.7 (42.1)
⫺292.2 (117.1)
53.2 (1.6)
49.9 (1.3)
54.4 (0.8)
50.7 (1.4)
52.1 (2.1)
Anterior cortex
Normal strain
ε1 angle
553.2 (60.0)
556.6 (100.8)
⫺26.8 (0.9)
⫺14.8 (2.1)
441.4 (35.0)
517.1 (65.6)
⫺12.0 (2.0)
⫺17.9 (7.9)
2
⫺748.32 (77.9)
⫺759.6 (42.7)
⫺645.5 (188.0)
1
Values in parentheses for individual subjects are standard deviations for 10 separate stance phases. Grand mean standard deviation
is calculated from individual subject means.
2
Longitudinal strain (from single element aligned with long axis).
TABLE 2. Metatarsal midstance strain data (1.5 m/sec, 10 strides)1
Anterior cortex
Medial cortex
Lateral cortex
Subject
Normal strain
ε1 angle
Normal strain
ε1 angle
Normal strain
ε1 angle
574
539
616
Grand mean
⫺672.8 (37.6)
⫺1,291.51 (36.1)
⫺1,124.6 (91.4)
⫺1,029.6 (320.1)
⫺84.4 (1.2)
⫺23.5 (40.5)
77.9 (32.5)
⫺79.4 (34.7)
⫺8.43 (79.7)3
55
(1.9)
⫺35.8 (3.3)
73.1 (6.1)
30.83 (58.4)3
42.8 (29.0)
149.8 (19.4)
171.6 (38.2)
121.4 (69.0)
⫺30.9 (0.6)
⫺19.2 (2.5)
⫺23.9 (1.0)
⫺24.7 (5.9)
⫺89.2 (0.2)
⫺87.6 (4.5)
1
Values in parentheses for individual subjects are standard deviations for 10 separate stance phases. Grand mean standard deviation
is calculated from individual subject means.
2
Longitudinal strain (from single element aligned with long axis).
3
High standard deviations for this grand mean are a result of different gauge position relative to NA in animal 539 than in 574 and
616.
to the NA projected through the area centroid. The
polar moment of area, JN, is calculated as IN ⫹ INy.
NIH Image was also used to measure the diameters
of the bone in the A-P (Da-p) and M-L (Dm-l) planes,
and the perpendicular distance from the NA to the
locations on the periosteal cortex of maximum tension (at) and compression (ac) in the A-P plane.
These measurements permit calculation of the two
section moduli: ZNt, the section modulus of tension,
defined as IN ● at⫺1; and ZNc, the section modulus of
compression, defined as IN ● ac⫺1.
RESULTS
Table 1 presents normal strains and angles at
midstance for 10 typical strides at 1.5 m/sec for the
tibia; Table 2 presents the same data for the metatarsal. Calculated cross-sectional distributions of
strains at midstance are graphed in Figure 3. In
several instances (noted in Tables 1 and 2), not all
gauge elements were working, but normal strains
were used from the element aligned with the long
axis (no calculations of ⑀1° are possible from these
gauges). Normal strains from all metatarsal sites
were available from three animals; in one metatarsal (#539), the medial gauge was located on the
tensile side of the NA, whereas in the other two
individuals, the medial gauge was located more caudally, on the compressive side of the NA. Unfortunately, only one animal (#600) had all tibial gauges
working simultaneously. However, of the five animals studied, at least three gauges provided data
from each tibial site, and results are highly comparable between individuals (Table 1). We therefore
calculated mean normal strains and angles for an
average tibia and for an average metatarsal, and
solved for the cross-sectional distribution of strains
using representative cross sections (#600 for the
tibia; #539 for the metatarsal), as shown in Figure 3.
As is evident from Figure 3 and Table 1, at midstance, both the metatarsal and the tibia have
higher compressive than tensile strains, which is
what one would expect from a loading regime that
combines bending with axial compression. Relative
to the (assumed vertical) ground reaction force in
the sagittal plane, the mean orientation of the tibia
at midstance is 29 ⫾ 4.5° (proximal end angled anteriorly), and the mean orientation of the metatarsal
at midstance is 14 ⫾ 2.7° (proximal end angled posteriorly). The average total strain (absolute sum of
strain on the anterior and posterior cortices) at midstance is 1,162 ⫾ 122 ␮⑀ in the tibia, and 1,850 ⫾
132 ␮⑀ in the metatarsal. The NA is less then 10°
from a mediolateral axis in both bones, and the
metatarsal is bent anteriorly concave, and the tibia
is bent posteriorly concave.
The orientation of principal tensile strains measured at gauges on the cortices at midstance corresponds reasonably well to a regime of bending. In
bending, ⑀1° of gauges on cortices subject to net
tension should be aligned with the long axis of the
bone, and ⑀1° of gauges on cortices subject to net
compression should be perpendicular to the long
axis of the bone. As Table 1 indicates, maximum
principal strains on the posterior and anterior cortices of the tibia are mostly aligned within 20° of
expected orientations (the anterior maximum principal strain is predicted to be 0°, and the posterior
maximum principal strain is predicted to be 90°,
LONG BONE LOADING
163
Fig. 3. Strain distribution in midshaft cross sections of sheep metatarsal and tibia at midstance at 1.5 m/sec. In all midshafts,
anterior (cranial) is at top, and lateral is to left. A: Metatarsal sheep 539. B: Mean metatarsal (calculated from Table 2). C: Tibia sheep
600. D: Mean metatarsal (calculated from grand means in Table 1). NA, neutral axis. Tensile strains are positive; compressive strains
are negative. X, location of strain gauges (not shown for B and D).
relative to the long axis of the bone). Similarly, the
maximum principal strain orientation on the anterior cortex of the metatarsal is within 10° of the
expected 90°. Principal strain trajectories cross the
neutral axis of bending at 45° angles, and principal
strains on the lateral and medial cortices should
therefore be close to this angle. The angle on the
medial cortex of the tibia is within 10° of this value
(Table 1). The angles on the medial and lateral cortices of the metatarsal are more variable (Table 2),
which may reflect less consistency in gauge position
relative to the neutral axis of bending.
Figures 4 and 5 illustrate the cross-sectional distribution of strains at different times during the
stance phase for the tibia and metatarsal. Figures 4
and 5 show that midstance is reasonably represen-
tative of strains at 25% through stance, but substantially different from the cross-sectional distribution
of strains shortly after touchdown and at 75%
through the stance phase. In particular, there is a
counterclockwise rotation and caudal migration of
the NA in both the tibia and the metatarsal between
touchdown and 75% of stance. In the tibia, the NA
rotates about 65°; in the metatarsal, the NA rotates
about 40°. In addition, as one would expect, strains
in both bones are quite low at touchdown, rise
quickly, and then decline again after midstance.
Figure 6 compares the orientations of the experimentally determined neutral axes of bending with
the orientation of the principal centroidal axes
around which the cross sections offer the greatest
and smallest resistance in bending in the three
164
D.E. LIEBERMAN ET AL.
Fig. 4. Strain isoclines in 100-␮⑀ intervals in tibia of sheep
600 at 1.5 m/sec at different points during stance phase. NA,
neutral axis. Anterior (cranial) is at top; lateral is to left.
Fig. 5. Strain isoclines in 100-␮⑀ intervals in metatarsal of
sheep 539 at 1.5 m/sec at different points during stance phase.
NA, neutral axis. Anterior (cranial) is at top; lateral is to left.
165
LONG BONE LOADING
Fig. 6. Orientation of neutral axis of bending (NA) calculated at midstance at 1.5 m/sec from strain data relative to orientation of
principal centroidal axes (CAmax and CAmin) around which cross section offers greatest resistance to bending relative to NA. Anterior
(cranial) is at top; lateral is to left. A: Sheep 539 metatarsal. B: Sheep 574 metatarsal. C: Sheep 616 metatarsal. D: Sheep 600 tibia.
TABLE 3. Cross-sectional properties (CSPs) calculated around experimentally determined
neutral axes vs. centroidal axes (% difference in parentheses)1
CSPs around experimentally determined NAs
Individual (bone)
539
574
616
600
(metatarsal)
(metatarsal)
(metatarsal)
(tibia)
IN
INy
1,025.3
768.6
1,001.9
951.8
1,234.3
828.7
1,415.3 1,594.0
CSPs (around axis through centroid)
JN
ZNc
ZNt
IN/INy
at/ac
Theta2
1,793.9
1,953.7
2,063
3,009.5
130.3
143.7
158.2
192.6
275.6
300
322.3
345.2
1.3
1.1
1.5
0.9
2.1
2.1
2
1.8
9
4
8
3
Ixx
Iyy
Imax
Imin
738.8 (28) 760 777.4 720.9
732.8 (27) 949 950.7 731.5
760.4 (38) 764 823.4 701.4
1,093.1 (23) 1,594 1,606.5 1,080.0
JC
ZC
Ixx/Iyy
1,498.3 (16)
1,682.2 (14)
1,524.7 (26)
2,686.7 (11)
65.4 (50)
71.1 (51)
70.4 (55)
95.5 (50)
1.0 (23)
0.8 (27)
1.0 (33)
0.7 (22)
Difference ⫽ (CSP calculated from centroidal axis ⫺ CSP calculated from NA)*100. IN, SMA around experimentally determined NA;
INy, SMA perpendicular to experimentally determined NA, through area centroid; JN, polar moment of area based on experimentally
determined axes (IN, ⫹ INy); ZNc, section modulus of compression around experimentally determined NA; ZN, section modulus of
tension around experimentally determined NA; ac/ac, ratio of distance from experimentally determined NA to cortices subject to
tension (at) and compression (ac); Ixx, SMA around mediolateral axis through centroid; Iyy, SMA around anteroposterior axis through
centroid; Imax, maximum SMA around a centroidal axis; Imin, minimum SMA around a centroidal axis (perpendicular to Imax); Jc,
centroidal polar moment of area (Ixx ⫹ Iyy); Zc, centroidal section modulus. See text for further details.
2 °
IN counterlockwise from horizontal or mediolateral (Ixx).
1
metatarsals and one tibia for which a complete set of
strain data was available (see Table 1). Table 3
summarizes cross-sectional properties, and differences between area properties calculated using the
experimentally determined bending axes and the
centroidal axes determined solely from the geometry
of the cross sections. These results show that the
metatarsal and tibia are subjected primarily to
bending in near-sagittal planes (the angle of the
neutral axis of bending relative to the mediolateral
plane, ⭋, is less than 10° for all midshafts at mid-
stance). The NA at midstance does not run through
the centroid (indicated by the intersection of the
CAmax and CAmin axes) of either the metatarsal or
the tibia, but is shifted about two-thirds of the way
towards the caudal (posterior) cortex in the metatarsal, and about two-thirds of the way towards the
cranial (anterior) cortex in the tibia, most likely due
to the superimposition of axial compression on a
bending regime (see above).
The orientation of the neutral axis does not correspond with the maximum principal centroidal axis
166
D.E. LIEBERMAN ET AL.
TABLE 4. Least square regression of logged cross-sectional properties calculated using estimated NA vs. CA
Independent
variable1
IN
IN
ZNt
ZNt
ZNc
ZNc
JN
JN
1
2
Dependent
variable2
N
Bone
Slope
Slope 95%
confidence interval
Intercept
Intercept 95%
confidence interval
r
Ixx
Ixx
Zc
Zc
Zc
Zc
Jc
Jc
30
30
30
30
30
30
30
30
Metatarsal
Tibia
Metatarsal
Tibia
Metatarsal
Tibia
Metatarsal
Tibia
0.98
1.06
0.94
0.99
0.94
0.99
1.02
1.05
0.87–1.09
0.91–1.21
0.80–1.08
0.79–1.20
0.80–1.08
0.79–1.20
0.96–1.08
0.99–1.12
⫺0.13
⫺0.70
2.12
0.95
2.62
⫺0.65
⫺0.27
⫺0.58
⫺0.97–0.71
⫺1.88–0.45
1.27–2.98
⫺1.09–1.28
1.84–3.41
⫺2.00–0.69
⫺0.76–0.21
⫺1.16–0.10
0.94
0.94
0.91
0.87
0.91
0.87
0.99
0.98
Calculated using NA derived from at/ac from Table 3.
Calculated using CA.
of the midshaft cross sections. Instead, it falls close
to the minimum principal centroidal axis of the tibia
cross section, and runs at approximately 45° angles
to the centroidal axis of the metatarsal cross sections (Fig. 6). Imax therefore does not predict the
plane of bending at midstance. Because of the
roughly tubular shape of the metatarsal, absolute
values of Imax and Imin are fairly similar in magnitude, but orientations of CAmax (around which Imax
is calculated) are between 38 – 45° from the observed
plane of bending (NA). In the tibia, the nearly A-P
orientation of CAmax indicates that the midshaft is
strongest at resisting mediolateral bending, and
CAmax thus is nearly perpendicular to the experimentally determined NA.
Because of the differences between principal centroidal axes and experimentallydetermined neutral
axes, calculations of Ixx in which the NA passes
through the centroid and runs strictly mediolaterally are highly inaccurate in terms of their absolute
value relative to IN (the second moment of area
calculated around the experimentally determined
neutral axis), ranging in error from 23–38% (Table
3). Values of J are better (differences range from
11–26%), but it should be noted that INy (which
contributes to JN) was calculated somewhat arbitrarily by assuming that the NA passed through the
centroid, perpendicular to IN. This assumption in
calculating INy reduces errors in J depending on ⭋,
the orientation of IN relative to the mediolateral
axis. As ⭋ approaches 0°, INy and Iyy will be identical. Finally, values of Z (section moduli) are extremely divergent, depending on whether they were
calculated using NA vs. CA (differences can be up to
55%), because errors in the second moments of area
are magnified by the inclusion of distances from the
NA to the outer cortices of compression and tension
(ac, at). As Table 3 shows, the ratio of at/ac is about
2.0 in the metatarsals and 1.8 in the tibia, both
different from 1.0, the value this ratio assumes
when Z is calculated around a centroidal axis.
Finally, Table 4 compares the slopes and intercepts of regressions for logged values of I, J, and Z in
a large sample of juvenile, subadult, and adult sheep
in which these parameters were calculated using the
area centroid (Ixx, Jc, and Zc) as well as the average
position of the NA relative to the anterior and pos-
terior cortices (at/ac from Table 3) from the straingauged animals (IN, JN, ZNc, and ZNt). We caution
that these regressions involve several assumptions,
most notably that the position of NA determined in
the small sample of strain-gauged sheep (summarized in Table 3) can be used to estimate the position
of NA in a larger sample of sheep, most of which are
older. This assumption needs to be tested. In addition, because we calculate NA position from the
mean ratio of at/ac, we are calculating cross-sectional properties using an anatomical plane rather
than experimental bending planes, which probably
vary in location between individuals. Thus correlations between the two ways of calculating crosssectional properties are undoubtedly inflated. With
these caveats in mind, the correlations between
logged cross-sectional properties calculated using
CAs vs. NAs are high, with no slopes that differ
significantly from 1.0, but with widely varying intercepts. As one would predict, calculations of crosssectional properties using CAs underestimate to different degrees the values based on experimentally
derived estimates of NA position.
DISCUSSION
The results of this study lend support to previous
models of limb-loading that predict or show long
bone midshafts to be strained by a combination of
bending and axial compression, with the neutral
axes of bending displaced towards the cortex subject
to tension (e.g., Pauwels, 1948; Lovejoy et al., 1976;
Biewener et al., 1983, 1988; Rubin and Lanyon,
1982; Biewener and Taylor, 1986, Gautier et al.,
2000; Gies and Carter, 1982; Gross et al., 1992). In
the sheep metatarsals and tibiae studied here, compressive strains are higher than tensile strains on
opposing anterior and posterior midshaft cortices,
with NAs shifted away from the centroid and towards the cortex, subject to tension in both midshafts. The angles of principal tensile strains relative to the long axes of the bones also suggest that
bending is the predominant loading regime to which
the bones were subjected.
The first hypothesis (that cross-sectional geometric properties calculated around an experimentally
determined NA are equal to or approximate those
calculated through the CA) is rejected in this study.
LONG BONE LOADING
Because most estimates of bending rigidity (I, J, and
Z) depend on the squared distance of unit areas from
NA, slight deviations in the position and orientation
of the neutral axis relative to the centroidal axis will
have large effects on the absolute values of crosssectional properties. Thus, the shift in NA position
relative to the centroid, along with differences in NA
orientation relative to the orientation of Imax documented in the sheep tibia and metatarsal, introduce
substantial differences between cross-sectional geometric properties calculated using CA vs. NA (Table
3). Differences in I range from 23–38%. Estimates of
Z in the anteroposterior plane have even higher
differences (50 –55%), because they compound differences in Ix along with errors in the bending moment
arm (assumed to be half the diameter of the bone in
the plane of bending). Differences in J (IN ⫹ INy
compared to Ixx ⫹ Iyy) are somewhat lower (11–26%),
largely because the way in which we calculated INy
gives broadly comparable results to Iyy, assuming
NA to run through the cross-sectional centroid. Finally, descriptors of cross-sectional shape (Ix ● Iy⫺1
vs. IN ● INy⫺1) also have large differences (23–33%).
While the absolute value of cross-sectional properties such as I, J, and Z tends to be quite different
from the value of the same parameters calculated
from experimentally determined NA position, the
pattern of differences between the two sets of parameters is highly correlated. As shown in Table 4, when
data on midstance NA position from strain-gauged
animals are extrapolated to a larger sample, the
correlations between logged cross-sectional parameters measured using NAs vs. CAs are high (r2 ⱖ
0.87), and none differ significantly from 1.0. Crosssectional parameters calculated using CAs tend to
underestimate the same parameters calculated using NAs, but to different degrees. Further research
from a larger sample is necessary to evaluate more
completely the relationship between cross-sectional
geometric properties determined using centroidal
vs. experimentally determined NAs.
The second hypothesis tested here (that the NA
coincides in its orientation with the principal, centroidal axis of the cross section, i.e., CAmax, the axis
around which the cross section offers greatest resistance to bending) is also rejected. As documented
here, the tibia and metatarsal are predominantly
bent in sagittal planes around mediolateral neutral
axes at midstance, yet the maximum resistance to
bending is offered in diagonal or mediolateral planes
for the metatarsal and tibia, respectively. These results corroborate some previously reported experimental studies of cross-sectional strains and crosssectional geometry in mammals and birds, which
found that bones were not necessarily most reinforced structurally in the direction in which peak
bending strains occur (Rubin, 1984; Gross et al.,
1992; Cubo and Casinos, 1998), and that the orientation of CAmax does not necessarily correspond with
experimentally determined directions of peak bone
bending during all locomotor activities (Demes et al.,
167
1998, 2001). Midshaft cross-sectional shapes of
limbs therefore do not necessarily reflect habitual
loading patterns during locomotion in the same
ways in all animals.
Variations in bone design as well as speed and
gait certainly complicate any effort to interpret the
relationship between orientations of peak bending
and cross-sectional geometric properties. As noted
above, one study (Szivek et al., 1992) found the
canine femur to be subjected to higher anteroposterior bending at faster speeds, which appears to correlate with the slightly anteroposteriorly elongated
shape of the femoral midshaft. The one other study
that explicitly compared midshaft strains in walking
and galloping macaques to midshaft cross-sectional
shape found a slightly improved correspondence between the orientations of CAmax and NA at higher
speeds for the tibia, but not the ulna (Demes et al.,
1998, 2001). These contrasting results are hard to
interpret, and more experimental data are required
to understand the relationship between bone loading and cross-sectional shape. One hypothesis is
that distal bone midshafts are optimized to resist
more variable loading patterns because they tend to
have lower safety factors. For example, in the sheep
studied here, the metatarsals are approximately tubular, causing the bones to have similar resistance
to bending in most axes (Imax and Imin are almost
equal). However, the same explanation does not apply to the horse third metacarpal, whose wide shape
gives it greatest strength around an anteroposterior
axis in spite of huge strains (⫺2,400 ␮⑀ of compression, and 610 ␮⑀ of tension) around a mediolateral
axis (Gross et al., 1992). Accordingly, some researchers suggest that certain bones sacrifice strength in
particular planes order to increase the predictability
of loads in response to a variety of loading conditions
(Lanyon, 1984; Bertram and Biewener, 1988), a
problem that may be especially important in distal
elements. Alternatively or additionally, midshaft
bone shape may reflect constraints on muscle attachments. In the case of the sheep tibia, the wide
shape may have little to do with resistance to bending, but instead results from the need to accommodate a sufficiently large surface area for the origins
of the major ankle flexors and extensors. More experimental data are needed to test these hypotheses,
but on the basis of the available evidence, it is most
sensible to conclude that cross-sectional properties
may not provide reliable information about patterns
or magnitudes of habitual loading. Consequently, we
caution against attempts to infer habitual patterns
of loading based on analyses of cross-sectional geometry (including Imax ● Imin⫺1) in the absence of experimental data for the bone and species in question.
The third hypothesis tested here (that the orientation and position of NA during the stance phase
are stable and similar to its position during midstance) is partially rejected. During stance phase in
the sheep, there is a substantial rotation (40 – 65°)
168
D.E. LIEBERMAN ET AL.
along with a slight posterior translation of the NA in
both the metatarsal and tibia. As noted above, these
findings indicate more change in NA position than
reported in other studies on the horse metacarpal
(Gross et al., 1992), the dog femur (Szivek et al.,
1992), and the macaque ulna and tibia (Demes et al.,
1998, 2001). Therefore, NA position may be more
variable than previously hypothesized, depending
on element and species. However, in terms of the
magnitude of bending strains across stance, the results of this study are consistent with other studies.
In the sheep, strain magnitudes are low at touchdown, reach near peak levels at 25% through stance
phase, peak at midstance, and then decline substantially by 75% through stance phase. Consequently,
the magnitude and orientation of loading at the midshaft are somewhat dynamic, but consistent through
that part of stance when strains are high. The forces
that cause these changes need to be studied but
probably include shifts in the orientation of the
ground and joint reaction forces relative to limb
segment orientation, combined with changes in
muscle activity.
We therefore conclude that cross-sectional properties estimated using centroidal axes have two major
problems that need to be considered explicitly when
applied to humans and other species in which in vivo
strain data are not available. First, while it has long
been appreciated the superimposition of bending
and axial compression shifts NA away from the
cross-sectional area centroid of most bones, the effect of this shift on quantitative estimates of bone
strength had not been previously measured. Our
study shows that calculations of cross-sectional resistance to bending around a centroidal axis differ
substantially in absolute value from calculations of
the actual cross-sectional resistance to bending in a
bone around the neutral axis. Second, the results of
this study suggest that there is no consistent and
predictable relationship between the shape of a bone
midshaft and the nature of the loading regime it
encounters.
Before we turn to the implications of these findings for comparative studies of cross-sectional geometry in humans and other mammals, several additional issues merit discussion. Perhaps the most
important is what strain gauges measure in treadmill studies vs. what bones actually experience on
more normal, varied terrains. As described above,
the methodology we used here calculates the position of the NA and the isoclines of strain away from
the NA, based on the relative magnitude of strains
normal to the midshaft cross section at three locations. Thus the pattern of relative strain between
the gauges determines NA position, while the absolute magnitude of strains determines the isocline
gradient. It follows that the position of NA provides
information solely on how the bone is loaded, while
the magnitude of strains provides information on
how well the bone is counteracting the loads. Obviously, animals do not usually run on treadmills, and
the nature of loads to which their limb bones are
subjected is certainly much more varied in terms of
the orientations of ground reaction forces, the orientations of limb elements relative to the substrate,
and the force produced by different muscles. While
some subtle kinematic and electromyographic differences were observed between treadmill and overground locomotion, studies on humans (Elliott and
Blanksby, 1976; Scardina et al., 1985; Arsenault et
al., 1986; Wank et al., 1998; Nigg et al., 1995; Alton
et al., 1998; White et al., 1998) and various quadrupeds (Wetzel et al., 1975; Buchner et al., 1994; Barrey et al., 1995) show that treadmill walking and
running are reasonably representative of overground gaits. However, the movement spectrum of
most species exceeds the range of movements that
can be elicited on a treadmill. The strains we measure on a treadmill therefore do not reflect the total
range of loading patterns, but probably reflect a
reasonable approximation of the typical loading pattern.
An additional issue is the limitation of sheep or
any other animal model for making inferences about
different animals, including humans. Sheep have
many obvious differences from humans, including
less muscled distal limb elements, three major long
bones per limb, and gaits that are constrained to
sagittal movements. But the beam model this study
tested in sheep is the same as the beam model applied to humans and other species. Thus, if the application of the beam model tested here does not
hold for sheep (which should be more predictable
than humans), then our null hypothesis should be
that the same assumptions do not hold for humans
or other animals until proven otherwise.
Implications
In conclusion, we wish to highlight four major
implications of these results for analyses of crosssectional properties in the absence of strain data.
First, we do not suggest that researchers abandon
using cross-sectional geometry as a means of comparing bones between individuals and populations,
or even between species. Analyses of cross-sectional
geometry still provide better information about long
bone function than simple measures of cross-sectional area. However, the results here confirm what
was long suspected: that not all cross-sectional parameters are equally useful, and none are likely to
be accurate in terms of absolute value. Calculations
of I, J, and Z based on neutral vs. centroidal axes are
highly correlated, but slopes and intercepts between
the two sets of parameters will vary considerably,
depending on a variety of geometrical and mechanical factors noted above. Thus, analyses that compare cross-sectional properties for the same bone
within a species, or between very similar species in
terms of musculoskeletal design and function (e.g.,
goats and sheep) where one can assume that patterns of bending are comparable, should yield results whose patterns are correct, but whose absolute
169
LONG BONE LOADING
values are likely to be incorrect. For example, comparisons of midshaft J estimated from CAs in the
femur between human populations (e.g., Ruff and
Larsen, 1990), or between playing and nonplaying
side arms of tennis players (e.g., Ruff et al., 1993,
1994), would differ from J calculated using actual
NAs (whatever they may be) mostly in terms of
absolute values, but the relative comparisons would
probably be robust. In contrast, analyses that compare cross-sectional geometry across species with
substantial differences in kinematics and anatomy
(e.g., Polk et al., 2000) may be more difficult to
interpret, even after standardizing by body mass
and element length. While such studies often separate animals by locomotor group (e.g., Ruff and Runestad, 1992; Polk et al., 2000), more research is
necessary to determine the biomechanical consequences or bases of observed differences in both
absolute and relative values of cross-sectional parameters.
A second implication of these results is that J is
probably the best single SMA parameter to use for
analyses of cross-sectional geometry in the absence
of experimental data on element loading. In typical
cross sections, J provides the most accurate estimate
of average bending rigidity, because errors in determination of NA position (and hence IN) are offset by
errors in the orthogonal second moment area, INy,
with the caveat that INy errors will increase as the
angle of Ix deviates from 0°. Note, however, that J is
likely to produce erroneous estimates of resistance
to torsion in noncircular cross sections (Wainright et
al., 1976; Daegling, 2002). In addition, the section
modulus, Z, is probably the worst SMA parameter to
use in the absence of experimental data, because it
compounds errors in Ix with errors in the distance
from NA to the cortex of bending.
Third, we suggest that interpretations of function
or habitual loading history based on cross-sectional
shape are problematic in the absence of in vivo data
on how bones are actually deformed. Evidently, and
for reasons we have yet to understand, the many
complex influences that contribute to bone shape
pose serious challenges for efforts to interpret variations in cross-sectional geometric properties. For
example, there has been a longstanding discussion
about the meaning of differences in bone-shaft shape
between modern humans and Neanderthals, the latter of whom typically have more bone mass distributed along the mediolateral axis of the femoral midshaft (Trinkaus, 1997; Trinkaus and Ruff, 1999;
Trinkaus et al., 1999). Such differences are hard to
interpret mechanically. Trinkaus (1986) suggested
that the relatively wider Neanderthal midshaft
might reflect more resistance to forces created by
lateral movement during activities by archaic than
by modern humans. In addition, Trinkaus et al.
(1998a, p. 5,840) suggested that such differences in
midshaft shape may be “evidence of a shift in locomotor patterns, with the presence of the femoral
anteroposterior reinforcement associated with in-
creased mobility and seen more frequently among
early modern humans than among the Neandertals.” However, as noted by Ruff (1995) and
Trinkaus and Ruff (1999), differences in pelvic
shape (Rak and Arensburg, 1987) or other anatomical factors unrelated to behavior may also contribute to the observed differences in Imax ● Imin⫺1 between these species. We therefore agree that, at this
point, it is not possible to infer reliably what, if
anything, such differences in midshaft shape mean.
Finally, these data raise most clearly the problem
that we need a better understanding of how bones
are loaded, and the relationship between bone crosssectional shape and both peak and habitual strains.
Analyses of cross-sectional geometry do provide new
insights, not possible from analyses of cross-sectional area, and we encourage their use. But the lack
of any simple, predictable relationships between
bone function and midshaft shape complicates interpretations of shape, and potentially invalidates comparisons between species. Further research is
needed to understand better how orientations of
limb elements relative to ground reaction forces and
local muscle forces influence patterns and magnitudes of bone shaft loading not only during treadmill
running, but also during normal activities, when
variability in strain patterns will be even greater
(Demes et al., 2001).
ACKNOWLEDGMENTS
We thank A. Biewener, S. Cote, M. Devlin, F. Guy,
S. Martin, O. Pearson, H. Pontzer, S. Psutka, P.
Ramirez, and F. Yates for technical assistance
and/or helpful discussions. Useful comments were
provided by the reviewers.
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